Question

which equation is equivalent to \\(\sqrt{x^2 + 81} = x + 10\\)?
options:
\\(x + 9 = x^2 + 20x + 100\\)
\\(x + 9 = x + 10\\)
\\(x^2 + 81 = x^2 + 100\\)
\\(x^2 + 81 = x^2 + 20x + 100\\)

Explanation:

Step1: Square both sides of the equation

To eliminate the square root, we square both sides of the equation \(\sqrt{x^2 + 81}=x + 10\). Squaring the left side gives \(x^2+81\) (since \((\sqrt{a})^2=a\) for \(a\geq0\)), and squaring the right side gives \((x + 10)^2\).

Step2: Expand the right - hand side

We use the formula \((a + b)^2=a^2+2ab + b^2\) to expand \((x + 10)^2\). Here, \(a = x\) and \(b = 10\), so \((x + 10)^2=x^2+2\times x\times10+10^2=x^2 + 20x+100\).
So the equation becomes \(x^2+81=x^2 + 20x+100\).

Answer:

\(x^{2}+81=x^{2}+20x + 100\) (the left - most option among the four vertical options)